dGMW function

The Generalized modified Weibull distribution

Density, distribution function, quantile function, random generation and hazard function for the generalized modified weibull distribution with parameters mu, sigma, nu and tau.

dGMW(x, mu, sigma, nu, tau, log = FALSE)

pGMW(q, mu, sigma, nu, tau, lower.tail = TRUE, log.p = FALSE)

qGMW(p, mu, sigma, nu, tau, lower.tail = TRUE, log.p = FALSE)

rGMW(n, mu, sigma, nu, tau)

hGMW(x, mu, sigma, nu, tau, log = FALSE)

Arguments

• x, q: vector of quantiles.
• mu: scale parameter.
• sigma: shape parameter.
• nu: shape parameter.
• tau: acceleration parameter.
• log, log.p: logical; if TRUE, probabilities p are given as log(p).
• lower.tail: logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
• p: vector of probabilities.
• n: number of observations.

Returns

dGMW gives the density, pGMW gives the distribution function, qGMW gives the quantile function, rGMW

generates random deviates and hGMW gives the hazard function.

Details

The generalized modified weibull with parameters mu, sigma, nu and tau has density given by

c("$f(x)= \\mu \\sigma x^{\\nu - 1}(\\nu + \\tau x) \\exp(\\tau x - \\mu x^{\\nu} e^{\\tau x})\n$", "$[1 - \\exp(- \\mu x^{\\nu} e^{\\tau x})]^{\\sigma-1},$")

for x>0.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function
curve(dGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, to=0.8,
      ylim=c(0, 2), col="red", las=1, ylab="f(x)") 
 
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5),
      from=0, to=1.2, col="red", las=1, ylab="F(x)")
curve(pGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5, lower.tail=FALSE),
      from=0, to=1.2, col="red", las=1, ylab="R(x)")

## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qGMW(p, mu=2, sigma=0.5, nu=2, tau=1.5), y=p, xlab="Quantile",
    las=1, ylab="Probability")
curve(pGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, add=TRUE, col="red")
 
## The random function
hist(rGMW(n=1000, mu=2, sigma=0.5, nu=2,tau=1.5), freq=FALSE,
    xlab="x", main ="", las=1)
curve(dGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5),  from=0, add=TRUE, col="red")
 
## The Hazard function
par(mfrow=c(1,1))
curve(hGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, to=1, ylim=c(0, 16),
      col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters